Defining polynomial
\(x^{12} - 176 x^{9} + 8228 x^{6} + 90508 x^{3} + 58564\) |
Invariants
Base field: | $\Q_{11}$ |
Degree $d$: | $12$ |
Ramification exponent $e$: | $3$ |
Residue field degree $f$: | $4$ |
Discriminant exponent $c$: | $8$ |
Discriminant root field: | $\Q_{11}(\sqrt{2})$ |
Root number: | $1$ |
$\card{ \Aut(K/\Q_{ 11 }) }$: | $6$ |
This field is not Galois over $\Q_{11}.$ | |
Visible slopes: | None |
Intermediate fields
$\Q_{11}(\sqrt{2})$, 11.4.0.1, 11.6.4.2 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Unramified/totally ramified tower
Unramified subfield: | 11.4.0.1 $\cong \Q_{11}(t)$ where $t$ is a root of \( x^{4} + 8 x^{2} + 10 x + 2 \) |
Relative Eisenstein polynomial: | \( x^{3} + 11 t^{2} \) $\ \in\Q_{11}(t)[x]$ |
Ramification polygon
Residual polynomials: | $z^{2} + 3z + 3$ |
Associated inertia: | $1$ |
Indices of inseparability: | $[0]$ |
Invariants of the Galois closure
Galois group: | $C_3:C_{12}$ (as 12T19) |
Inertia group: | Intransitive group isomorphic to $C_3$ |
Wild inertia group: | $C_1$ |
Unramified degree: | $12$ |
Tame degree: | $3$ |
Wild slopes: | None |
Galois mean slope: | $2/3$ |
Galois splitting model: | $x^{12} - 3 x^{11} - 112 x^{10} + 135 x^{9} + 4708 x^{8} + 2405 x^{7} - 82903 x^{6} - 159417 x^{5} + 429601 x^{4} + 1548270 x^{3} + 1263093 x^{2} + 15394 x - 41348$ |